The categorical theory of self-similarity

Peter Hines

We demonstrate how the identity $N\otimes N \cong N$ in a monoidal
category allows us to construct a functor from the full subcategory generated by
$N$ and $\otimes$ to the endomorphism monoid of the object $N$. This provides a
categorical foundation for one-object
analogues of the symmetric monoidal categories used by J.-Y. Girard in his Geometry
of Interaction series of papers, and explicitly described in terms of inverse
semigroup theory in [6,11].

This functor also allows the construction of one-object analogues
of other categorical structures. We give the example of one-object
analogues of the categorical trace, and
compact closedness.
Finally, we demonstrate how the categorical theory of
self-similarity can be related to the algebraic theory (as presented in
[11]), and Girard's dynamical algebra,
by considering one-object analogues of projections and inclusions.